Construct a polynomial function with the following properties: third degree, 3 is a zero of multiplicity 1, −1 is the only other zero, leading coefficient is 4.

Question

College Algebra

Answer 1

Third degree means the highest power in the polynomial is x³

One thing before we get started. When a problem says that a polynomial has a zero, for example let's say it has zero x=2, then that means that the polynomial has a factor (x-2). To solve this problem, it's easier to talk about factors.

A third degree polynomial has either 1 or 3 real number factors. (If it only has 1, that means the other 2 are imaginary, but that's another topic.)

First, the problem says it has a zero x=3, That means it has a factor (x-3). Because it has multiplicity of 1, there is only one factor (x-3).

The problem says that the only other zero, x=-1. That means the polynomial has a factor (x+1). But, because there must be 3 factors, the third factor is also (x+1).

So the polynomial must be a(x-3)(x+1)(x+1), where a can be any number.

It says that the leading coefficient is 4. So, a=4.

So, the polynomial function is f(x) = 4(x-3)(x+1)(x+1) = 4x³ - 4x² - 20x - 4.